Antiferromagnetism at half-filling

I will now allow for antiferromagnetic ordering in the calculations. Figure 5.12 shows the resulting phase diagrams for t₂/t₁ = 0.6 (upper panel) and t₂/t₁ = 0.8 (lower panel) for different temperatures and interaction strengths. The phase diagrams were constructed by fitting the magnetization to the actually calculated data points. This of course means that the phase boundaries shown here must be considered as guess only. However, as one does not expect any strange structures to appear, this guess will presumably represent the true phase boundary within a few percent. The full lines in figure 5.12 are the PMIT transitions. Note that for both diagrams the same division of axes was chosen.

In contrast to the Hubbard model for a bipartite lattice with t₂ = 0, there now exists a finite critical value U_c^AF >0, below which no antiferromagnetism can be stabilized even for temperature T −→ 0. With increasing frustration the paramagnetic-antiferromagnetic transition is shifted towards higher inter-action strengths and lower temperatures, while the PMIT is shifted towards lower interactions strengths. So obviously the PMIT is shifted towards the phase boundaries of the antiferromagnetic dome. So far this is the expected effect of the NNN-hopping which introduces frustration to the antiferromag-netic exchange. However, note that although t₂/t₁ = 0.8 represents already a very strongly frustrated system, the PMIT still lies well covered within the antiferromagnetic phase.

I will now have a closer look at the paramagnetic-antiferromagnetic transition.

Here, R. Zitzler et al. [147] made the prediction that one has to expect a first order transition close to the critical U_c^AF at low temperatures; while at larger values of U again a second order transition was found. Figure 5.13 shows the staggered magnetization for different temperatures and interaction strengths at fixed t₂/t₁ = 0.8. The upper panel collects data for the transition at low temperatures at the lower edge of the antiferromagnetic phase. The full lines represent the transition from the paramagnetic to the antiferromagnetic state with increasing interaction strength for two different temperatures, while the dashed lines represent the transitions from the antiferromagnetic to the para-magnetic state with decreasing interaction strength. In the upper panel (low T) one can clearly see a hysteresis of the antiferromagnetic transition. This hysteresis as well as the jump in the magnetization are clear signs for a first or-der transition. This antiferromagnetic hysteresis is very pronounced for strong frustration but numerically not resolvable for example for t₂/t₁ = 0.2. The hysteresis regions seems to shrink with decreasing t₂ and eventually cannot be resolved anymore with numerical techniques. Note that such a hysteresis is also found in the two-sublattice fully frustrated model [147], which means that this is quite likely a generic effect in frustrated systems at intermediate coupling strengths. The lower panel in Fig. 5.13 shows the staggered magnetization for temperatures just below the corresponding N´eel-temperatures and at large interaction strengths. Here the magnetization vanishes smoothly, which is the behavior expected for a second order phase transition. In summary I thus find a first order transition at the critical interaction U_c^AF, where the

antiferro-5.4 Frustrated Bethe Lattice 83

Figure 5.12: The upper (lower) panel shows theT −U phase diagram for t2/t1= 0.6 (t2/t1 = 0.8). The colored area represents the antiferromagnetic phase, while the black area represents the paramagnetic phase. The lines show, where the PMIT in the paramagnetic phase would occur. The phase diagrams were constructed by fitting the magnetization to approximately 70 calculated data points. Additional calculations were performed to find the PMIT-lines.

0.8 1 1.2

increasing U at T/W=1.7e-4 decreasing U at T/W=1.7e-4 increasing U at T/W=1.4e-3 decreasing U at T/W=1.4e-3

1 1.5 2 2.5

Figure 5.13: Staggered magnetization versus interaction U/W for two different tem-peratures and t2/t1 = 0.8. In the upper panel there are two transition lines for each temperature, representing either increasing or decreasing interaction strength. The region between both lines embodies a hysteresis region. The lower panel shows the transition for large interaction strengths and high temperatures. Here no hysteresis region can be found, but a smooth transition. The lines are meant as guide to the eye.

magnetism sets in, and a second order transition for large Coulomb parameter U ≫ W. The merging from both transition lines is an interesting point in itself. There must be a critical point where the first order transition changes into a second order transition. It is however not possible to resolve this merging within DMFT/NRG. The magnetization of the system becomes very small in this region, so it is not possible to distinguish between a (tiny) jump and nu-merical artifacts of a smoothly vanishing order parameter. Consequently, one cannot decide anymore of what order the transition is.

Antiferromagnetic metallic states at half-filling were reported in earlier pub-lications [148, 165–167] and later ruled out again. In my calculations I saw no evidence for an antiferromagnetic metallic state at half-filling. Especially for strong frustration t₂/t₁ ≈ 0.8 the system jumps directly from a paramag-netic metallic solution into an antiferromagparamag-netic insulating solution with large magnetization. In the papers cited, the region showing an antiferromagnetic metallic solution broadens with increasingt₂. This prediction I clearly cannot confirm, as discussed above. Only in systems with small to intermediate frustra-tion there are narrow interacfrustra-tion regimes where I observe a small finite weight at the Fermi level. One must however consider that the occupation number is not exactly one but only within 0.5%, which influences the position of the gap.

It was also sometimes difficult to stabilize a DMFT solution in these regions. In summary, I cannot see any clear signs for an antiferromagnetic metallic state at half-filling in my calculations. If any exists, then only for rather low frustrations

5.4 Frustrated Bethe Lattice 85

in a very small regime about the critical interactionU_C^AF. To what extent these rather special conditions can then be considered as realistic for real materials is yet another question.